Forward Contract Hedging with Contingent Portfolio Programming

Transcription

1 Forward Conrac Hedging wih Coningen Porfolio Programming Ma-.08 Independen research projecs in applied mahemaics Oso Manninen, 60036T, Tfy s Augus 006

2 Conens Inroducion... Forward Conracs Coningen Porfolio Programming and How o Use I in Hedging Forward Conracs Coningen Porfolio Programming Forward conrac hedging wih CPP An Applicaion of Coningen Porfolio Programming in Porfolio Opimizaion Problem formulaion Case resuls and resul analysis Summary Model pros and cons Model resricions and usabiliy... 7 References...

3 Inroducion The purpose of his paper is o demonsrae how Coningen Porfolio Programming (CPP) can be used in managemen of he forward conrac porfolio when he underlining asse is a perishable commodiy, such as elecriciy, which canno be sored o be used laer. The decision problem is o choose an opimal porfolio of forward conracs for he delivery of commodiy ha has wo unknown variables, he spo price and he level of demand. The model compues he opimal levels of forward conracs o be bough in order o mee all he demand. The compued soluion can also be used as a sep by sep guide for choosing he opimal amoun of hedging for he remaining ime periods. Gusafsson and Salo (005) inroduced CPP which is applied in his paper in he forward conrac managemen. CPP is a model ha allows decision problems o be consruced ino a sae ree which can be solved wih linear programming ools. The paper presens he heoreical model and applies i on numerical examples. The numerical examples demonsrae how a problem of reasonable size can be solved wih he model. From he examples we can also see how he model akes ino accoun he decision maker s risk preferences by penalizing for variabiliy in he expeced reurn. The model has some resricions due o our assumpions. This paper ells some ways o enhance he model and provides some direcions for fuure research. The paper highlighs some imporan areas in order o develop applicaions based on he model. The paper is srucured as follows. Chaper inroduces he forward conracs. Chaper 3 inroduces he CPP model in brief and presens he forward conrac managemen problem and presen he way how CPP can be applied for i. Chaper 4 consiss of numerical examples and chaper 5 concludes wih a discussion.

4 Forward Conracs Forward conracs on commodiy are defined as conracs o purchase or sell given amoun of he commodiy a specific ime and a specific price (Luenberger, 998). Forward conracs are specified by legal documens which bind he conracors on he ransacion in fuure. The value of he forward conrac in elecriciy marke from one period o succeeding period is assumed o be he probabiliy adjused average of spo prices in he succeeding scenarios as elecriciy is non sorable. Thus, forward conrac value P (,' ) in period for delivery in period ( > ) ' is P n ( ' ) =, p i * S i, () i= where S i is he spo price in scenario i in period, p i is he probabiliy of ha spo price, and n is number of possible scenarios in period. Forward conracs are used when he decision maker (DM) wans o minimize he risk eiher on he availabiliy of a commodiy or on he price of ha commodiy in he fuure. This paper focuses on he laer, as our model requires he assumpion ha all demand is me. If we had o consider he availabiliy of he commodiy, we migh have o face a problem of no being able o mee he demand in all siuaions. Combining hese wo assumpions, ha all demand mus be me and ha he availabiliy of he commodiy could be resriced, would mean ha all demand mus be hedged for he final period. This is because here migh no be available commodiy o buy a he spo price, and hus he opimizaion problem would be o opimize he hedging for he final period (wihou he possibiliy o increase he supply wih buying a spo price). I migh also lead o resricions in available forward conracs. Basics on hedging wih fuures/forwards can be learn form Brealey and Myers (003). However, hey do no discuss he hedging of perishable commodiies as hey do no see simple link beween elecriciy s spo price and is fuure price (which is he case on many oher commodiies). Bu on some cases, as is in he Nordpool elecriciy marke, expecaions heory can be used o esimae he becoming spo price from exising fuure prices (see Luenberger, 998, for more informaion on expecaions heory). 3

5 This paper assumes ha he forward price can be derived from possible fuure spo prices using he equaion (). We assume ha a each momen he commodiy can be purchased in any size of unis boh a spo price and a forward conrac price. We also assume ha we are alking only abou perishable commodiies ha can no be purchased beforehand and hen sored o mee laer demand. Perishable commodiies differ from normal commodiies as hey can no be sored, which means ha hese commodiies mus be bough a he momen hey are needed or wih a previously signed forward conrac ha delivers he commodiy a ha specified momen i is needed. Furher we assume ha if he commodiy is no used or sold prior o expire period, i is wased (he forward conracs can no be sold a heir expiraion period because here would be issues relaing o finding buyer for i in insan and if periods would be longer hen here would be issues on pricing he forward conrac if some of i has already expired). We allow selling of forward conracs a any oher period (excep expiraion) on he price ha is equal o similar forward in ha period. For example, if we have bough forwards in period 0 o be delivered in period, we can sell hem in period a a price ha is equal o price of a forward from period o period. If he price would be differen, here would be an arbirage. 3 Coningen Porfolio Programming and How o Use I in Hedging Forward Conracs 3. Coningen Porfolio Programming The CPP model is defined by resource ypes, he sae ree and projec-specific decision rees (Gusafsson and Salo, 005). Resources are inpus and oupus ha are consumed or produced by he projecs. They can be producion facors (money, labour) or inangibles (inellecual propery). Sae ree represens he ime-sae model. Time horizon consiss of [0,...,T] periods and in each period here is a se of possible saes. The sae ree sars wih only one sae in he firs period. The nex period has already more saes. The predecessor in each sae can be defined recursively, hus if we know he beginning sae and he end sae we can define each sae in he beween. The sae ree is formed based on uncerain evens and heir probabiliies. Consequenly, he condiional 4

6 probabiliy of a cerain sae can be compued recursively using he condiional probabiliies. Projecs are inegraed ino he CPP via decision sequences. A decision sequence consiss of decision opporuniies. Thus can be formed a decision ree which consiss of he decision poins. In each decision poin he DM has knowledge abou he previous scenarios and he acions aken before wih regard o projec. I is assumed ha he decision poins form a consisen ree so ha each decision poin has a unique collecion of preceding decision poins. A projec managemen sraegy is defined by acion variables associaed wih decision poins in cerain projec. A porfolio managemen sraegy is he DMs complee plan of acion for all projecs in all saes. There is resource flow of cerain resources induced by he projec managemen sraegy. Acions influence he resource flow only in he curren sae (ha he acion is aken) and relevan fuure saes. The aggregae resource flow in cerain sae is obained by adding he resource flows of all projecs. CPP consiss of four se of consrains: decision consisency consrains, resource consrains, opional consrains and deviaion consrains. The CPP framework assumes ha he iniial resources are known and ha as acions are aken resources are eiher consumed or produced. The fuure amoun of resources can be hus compued in CPP a each poin as well as he resources in he erminal saes. The CPP model also akes ino accoun he risk profile of he DM and provides he opimizaion problem he pah ha he DM should follow a each decision poin. Wih he use of CPP projec porfolio selecion problem can be presened as a linear programming model. The soluion of he model suggess opimum decisions in each sae. In Table we have example probabiliies for a scenario ree. The probabiliies for he demand and price o move up or down independenly is 0.5 for boh and hus he probabiliy for cerain sae o follow is 0.5. Table. Example scenario Demand Up Down Price Up Down

7 Figure demonsraes his scenario ree for hree ime periods T = 0,,. From figure can easily be seen he uniqueness of each sae and is proceeding saes. Figure. Example sae ree T=0 T= T= T 3. Forward conrac hedging wih CPP In his secion he paper inroduces he principles which hen formulae forward conrac managemen wih CPP. This secion is based on a discussion paper by Salo (005). The decision problem is o choose an opimal porfolio for forward conracs for he delivery of commodiy, subjec o uncerainy abou (i) he spo price and (ii) he level of demand. The Planning horizon consiss of = 0,..., T periods. The spo price is assumed o follow a binomial laice where he probabiliy for going up is + p and he probabiliy for going down p subjec o he normalizaion condiion requiremen p + + p =. The muliplicaive upward change is u and muliplicaive downward change is d. The muliplicaive changes a he differen nodes are independenly disribued. The iniial spo price in he base scenario 0 s is ( 0) P (which is assumed o be known). The demand is assumed o follow a binomial laice where he probabiliy for upward shif 6

8 is + p and p similarly subjec o normalizaion condiion requiremen p + + p. The = muliplicaive changes are u for upward change and d for downward change. The iniial level of demand is D ( 0) (which also is assumed o be known). The se of scenarios describing he movemen of spo prices and level of demand is defined as S { s R s { 0, }, i =,, j,..., T} =. The elemens of hese marices are inerpreed so ha x ij = if he spo price goes up in period j, hen s ; oherwise he spo price goes down and we j = have s 0. Le us assume ha s S,. Now we can define he number of spo price moves j = + upwards unil period as q ( s ) = j = s j ; conversely we can define q ( s ) = = ( s ) j j consrucion we have q + ( s ) + q ( s ) =. This means ha spo price P ( s ) in scenario + q wrien as ( ) ( s ) q P s u d ( s ) ( 0) =. P. By s can be Likewise for he demand; if he demand in period j goes up, hen s ; oherwise he demand goes down and we have s 0. The backward operaion on scenarios is defined so ha if s S, >, hen ( ) by B ( s ) j = j = ' b s = s' S such ha s = s, j =,...,. All he predecessors of ij ij s are denoed. The dynamics and corresponding variables for he demand are defined in he same way as for he spo price. This means ha he number of imes demand moves up unil period is + ( s ) = = s j and he number of imes i has moved down is q ( s ) = ( s j ) q j q j = + ( s ) + q + q ( s ) =. The level of demand can be wrien ( ) ( s ) q D s u ( s ) d D( 0). Similarly =. Assuming ha he spo price movemens are independen from demand uncerainies implies ha he probabiliy of cerain scenario is + (, = is ( ) ( ) ) + qi s qi ( s ) p s = pi ( pi ) s,..., T i=. In each scenario s, here exis a forward conrac for he laer delivery '> a a price which is equivalen o he condiional expeced spo price for ha period, i.e. P ( ) = ' ' s, ' ' ' ' { ( )} p( s ) P( s ) s S s B s. () 7

9 The problem is o decide he amoun of forward conracs x( s ' ) 0, ', such ha he demand is saisfied and he exposure o risk is minimized (in he sense which is addressed below). If ' =, ' hen x( s, ' ) = x( s, ' ) is he amoun of commodiy ha is purchased a he spo price P ( s ). The key consrain sems from he requiremen ha all demand mus be saisfied wih he forward conracs or by purchasing wih he spo price, i.e., ' ( s, ' ) + x( s, ' ) D( s ) ' ( ) x =. (3) s B s The cash flows can be examined wih he help of resource surpluses (resource flows in CPP), derived from he iniial sock M >> 0. This means ha he resource surplus a he base scenario 0 0 s is RS( s ) M x( s 0,0) P( 0) = where he iniial sock M is an arificial variable ha can be employed o measure jus how much demand has been me. If s S, he corresponding surplus is RS ( s ) RS( s ) x( s, ) P( s,) x( s,) P( s ) =. (4) In he same way his can be generalized ino he expression RS i ' ' ( s ) = RS( b( s ) xi ( s, ) P( s, ) x( s, ) P( s ) ' s B ( s ). (5) The objecive funcion for he maximizaion problem is s S ( s ) RS( s ) EV = p. (6) T From he viewpoin of risk managemen, he quesion is how o accoun for he variabiliy and how o penalize for i. As in CPP, for each erminal scenario + ( s ) and ( s ) so ha s S T we define deviaional variables 8

10 + ( s ) EV ( s ) + ( s ) = 0 RS, (7) Where he erms + ( s ) 0 and ( s ) 0 scenario indicae (if posiive) wheher he resource posiion in s is larger or smaller han on he average. Now he objecive funcion can be augmened hrough an addiional erm which penalizes for variabiliy from he expecaion. The objecive funcion becomes s S + ( s )[ RS( s ) ρ( ( s ) + ( s )] max p, (8) T where he risk aversion coefficien ρ > 0 penalizes for variabiliy. 4 An Applicaion of Coningen Porfolio Programming in Porfolio Opimizaion 4. Problem formulaion The paper approaches he forward conrac managemen by focusing on hree simple examples. We assume ha he commodiy can no be sored and ha all he demand mus be me and can be purchased from he marke. This leads o a siuaion in which commodiies are eiher purchased beforehand wih he forward conracs, purchased a he concerning momen wih spo price or purchased wih some combinaion of hese wo previous alernaives. Le us consider a problem wih an iniial need of perishable commodiy D ( 0 ) = 00 and ha amoun varying by ime depending on various sources of uncerainy. A differen poins in imeline we have consumpion eiher increasing by u =. or degreasing by d = 0. 9 a cerain + probabiliies; = 0. p 6 for upward and p = 0. 4 for downward movemen. Similarly, iniial spo price is ( 0 ) = 00 P of and he price eiher rises by =. + u a probabiliy p = 0. 4 or decreases by 9

11 = 0.9 d a probabiliy of p = The ime horizon consiss of hree seps = 0,,, and he risk aversion coefficien ρ =. The iniial cash amoun is M = The problem is o deermine he opimal amoun of commodiy bough wih forward conracs in order o maximize he risk-adjused resource surplus in he erminal sae. The paper examines also he sensiiviy of he model in regards o he risk aversion coefficien ρ by rying hree differen coefficien values, ρ = 0, ρ = and ρ = 5. The values for price and demand were calculaed as well as he probabiliies of each sae in each period. The possible prices in he firs period are P ( s ) = ( 0 90) are D ( s ) = ( 0 90), while he possible demands. Now if we combine hese, we ge he following scenario marix for he firs period (see able. o see how he marix is compiled) 0,0 90,0 ( ) ( ) s, P s = 0,90 90,90 D, wih he following probabiliies ( ) s = p. Similarly hese can be calculaed for he second sae. Now he price marix is P ( s ) = ( ) and he demand is D ( s ) = ( ) ge he following marix. Combining hese we, 99, 99, 8,,99 99,99 99,99 8,99 D ( s ), P( s ) =,99 99,99 99,99 8,99,,8 99,8 99,8 8,8 0

12 wih he following probabiliies p ( s ) = Finally we need o calculae he fuure prices by using he equaion (). We use + o describe he cases when price has gone up in he period = and - o describe cases when price has gone down in he period =. P P P P ( 0,) = 98 ( 0,) = ( +,) = (,) = 88. In Figure, he whole example seup is presened. In each sae box (marked wih complee line borders), he P is for he spo price in ha sae and D is he demand. In probabiliy boxes (marked wih fragmenary line borders), p sands for probabiliy. Arrows ha go hrough probabiliy boxes represen possible following saes, and he probabiliy box ells he probabiliy for ha sae o occur. The probabiliy boxes on he lef side of sae boxes in period (marked wih fragmenary line wih dos borders) represen he probabiliy o evenually end o ha cerain sae. Finally, arrows wih fuure price ex on hem represen he possible forward conracs and heir prices.

13 Figure. Example scenario p=0.4 P = D = p= P =0 D =0 p=0.6 p=0.36 P = D =99 P =99 D = p= p= Fuure price 07.8 p=0.4 P =99 D =99 p= P 0 =00 D 0 =00 p=0.4 p=0.6 Fuure price 98 p=0.36 P 0 =0 D 0 =90 P =90 D =0 Fuure price p=0.4 p=0.6 p=0.36 p=0.4 p=0.4 p=0.6 p=0.36 P = D =99 P = D =8 P =99 D =99 P =99 D =8 P =99 D = P =99 D =99 P =8 D = p= p=0.056 p= p= p= p= p=0.96 p=0.4 Fuure price 88. p=0.4 P =8 D =99 p= p=0.4 P =99 D =99 p= P =90 D =90 p=0.6 p=0.36 P =99 D =8 P =8 D =99 p= p= Period T=0 Period T= p=0.4 P =8 D =8 p= Period T=

14 4. Case resuls and resul analysis This case was solved using normal Excel solver o maximize he arge funcion in order o demonsrae he usabiliy of he model. However, he model is no limied by he linear programming sofware ha is used. In he numerical case, he resuls were compued by using he equaion (5), from where we ge he ( s ) RS for each sae. Noice ha in period = 0 four saes, hus four ( s ) RS, and in period 0 we have one ( s ) = we have 6 RS ( s ) ge (by insering he iniial values presened in he chaper 4.) RS, in period = we have. Now using he formula we RS ( s 0 ) = *00 = For each sae in period =, he ( s ) RS is 0 0 ( s ) = P( s,) * x( s,) P( s )* x( s, ) RS, 3 0. period resource surplus where ( s ) fuures spo P is he price of ha sae in period = in ha period, ( s 0,) ( s 0,) 0 and x( s,) 0 is he spo demand of ha sae P is he fuure price in ha sae of he period = 0 o period = and x is he amoun bough wih hese fuure conracs in his sae of he period =. We also have he demand resricion, which says ha demand in each sae in period = mus be me. 0 Tha is x ( s, ) x( s,) = D( s ) For period = +., he ( s ) RS is similarly calculaed by using he equaion (3) 0 ( s ) = RS( s ) P( s,)* x( s,) P( s,)* x( s,) P( s )* x( s, ) RS, 3. period resource surplus fuures fuures spo 3

15 where, ( s ) P ( s 0,) and ( s,) RS is he resource surplus in he period = (depends of he sae in ha period), P are he fuure prices from corresponding periods o period = is he amoun fuures bough in period 0, x( s,) 0, x ( s 0,) 0 is he amoun bough in he period =, ( x s,) 0 is he amoun bough wih spo price and finally P ( s ) and ( s ) D are he spo price and demand in period = (boh hese depend on he sae in which he sysem is in period 0 = ). Again we have demand resricion ha says x ( s, ) + x( s,) + x( s,) = D( s ) Now by using he equaion (6) and ( s ) + we calculaed ( s ) and ( s ) RS we calculaed EV. Wih formula (6) and by using he EV,. Finally wih equaion (6) we formed he maximizaion problem. Calculaing he opimal hedging wih he iniial values presened in chaper 4. (and using ρ = ) we go ha he maximum is achieved when x( s 0,) = and x ( s 0,) = This means ha we should buy forwards in he period = 0 o be delivered in he period =. Now, for he second period, if we are in sae where price and demand have boh gone up, we would buy x ( s,) =. 0, which would mean ha we would hedge agains maximum possible demand in period = as we already have bough 8.8 forwards for period = in period = 0. For sae in which price has gone up and demand has gone down, resul is ha 4.55 of previously bough forwards (bough in he period = 0 o be delivered in period = ( s,) = ) would be sold and x of new forwards would be bough for he period =. Again he hedged amoun is he maximum possible demand in period =. For sae in which price has gone down and demand up, we would buy ( s,) =. 0 x and for price down demand down scenario he resul is ha 3.37 of previously bough forwards (bough in he period = 0 o be delivered in period = ) would be sold and ( s,) = x would be bough. Resul says ha for he period = minimum possible demand would be hedged and for he period = maximum possible demand would be hedged. The hedged forwards in he period = is a combinaion of forwards bough in periods = 0 and =. 4

16 Similarly he hedging can be calculaed for all he coefficiens. In he case ρ = 0, he amoun of hedging for he firs period can be anyhing beween zero and he minimum demand in he possible following sae, i.e. ( s 0,) = 90 possible demand in period = x. Also, in he period = 0 we would hedge for he minimum i.e. ( s 0,) = 8 x. In he period = we could hedge any amoun beween zero and he minimum possible demand in possible following sae in period =, so ( s, ) [ 0,min D( s )] x =, where is he sae in which he decision is made in period =. This is ' inuiive, as if we do no penalize for variabiliy, he only hing ha maers is he coss. And because he fuures are probabiliy weighed averages of possible prices, on average i would make no difference wha kind of porfolio we compose of buying spo or buying fuures, as long as we do no wase any excess capaciy (which is why we would only hedge agains he minimum possible demand in he nex period and res we would buy on spo.) In he case ρ = 5, resul is ha we will buy forward conracs for he maximum possible amoun of demand in he erminal period, i.e. x ( s,) 0 = 0 and ( s,) 0 = x. In he period = in saes where demand had gone down, fuures conracs are sold so ha fuures equal he maximum possible demand in he nex possible sae in he period =, i.e. forwards (bough in period = 0 for delivery in period = ) are sold. This is o adjus o he lower demand in he erminal period, and by selling he exra fuures we secure no losing any resources. This is also an inuiive resul, as if we are going o penalise grealy for he variabiliy, hen losing money wih buying oo much will be offse by no having any variabiliy in he final resource surpluses. As can be seen from he resuls, he model is very sensiive o changes in coefficien. Coefficien 0 means hedging o amoun of minimum demand and coefficien 5 means ha hedging will be o he maximum demand in he following periods. Furhermore, i can be noed ha here is no unique soluion in many scenarios; here can be found many combinaions beween he amoun of hedging and buying a a spo price. This is because of he uniqueness of each pah in he sae ree. I should be also noed, ha in our example we had only hree periods, and so only one forward ha was over longer ime period. If we had more ime periods, he number of hese over-a-periods forwards would increase, making he model slower o compue as well as harder o program. Of 5

17 course, using proper sofware wih more efficien algorihms could be used o diminish he problem. Bu using Excel has an advanage, as opimizaion programs are no available o everyone and easy for everyone o use. The resuls of he numerical examples are no more han illusraive as we had only hree ime seps. The compuaion ime was shor bu as he number of ime periods increase he compuing ime grows rapidly. Neverheless, Excel can be used o solve small problems. Gusafsson and Salo (005) also demonsrae ha larger problems can be solved using an opimizaion program. 5 Summary 5. Model pros and cons The model akes ino accoun he Decision maker s (DM) risk profile by allowing he DM o specify he risk aversion coefficien. However, he model penalized similarly excess capaciy and lack of capaciy. This could be enhanced so ha only excess capaciy is penalised. The model is scalable in he scale ha i is possible o compue in a reasonable ime. Togeher wih he fac ha ime periods are no bound o any lengh of ime inervals i is possible o use his model in any ime scale ha DM wans. The linear equaions and hus linear solving enable he model o be compued in a reasonable ime in many cases. Lineariy also means ha here is always a soluion for he problem and ha i is possible o solve. The model misses he cyclic characer of he markes. The model does no ake ino accoun he ime-series which of course affec he demand (which can be seen for example by observing he elecriciy demand in cerain imes of he year). However, we could consruc he sae ree by no using general formulas for prices and demands, bu insead values provided by expers ha ake also he cyclical characer ino consideraion. This would lead o he fac ha we canno 6

18 auomaically generae he ree, unless we would firs creae sofware o do ha. The model also misses he producion capabiliy of he DM. If he DM has is own producion capabiliy i can mee he demand also hrough own producion. This makes he decision making process much more complicaed. This may render he model less useful for some markes. Some limiaions of he CPP model are also included in he approach presened in his paper. The major resricion is he size of he decision ree as i grows exponenially when ime horizon is expanded. The sae ree limis he usabiliy of CPP. Even hough CPP is based on linear programming he model size grows grealy as he number of saes is increased or he ime horizon grows longer. Bu his is relaed o all ree based approaches and as Gusafsson and Salo (005) noe, he CPP model does have only a sligh percenage of variables and consrains compared o convenional decision ree. In addiion, Gusafsson and Salo (005) demonsrae ha CPP can be used in realisic sized problems o solve hem. CPP model has discree ime periods, which resrics he usage of he model in pracice. For example, he price of elecriciy on he markes consanly changes resuling in coninuous updaing of he calculaions. 5. Model resricions and usabiliy Forward conrac hedging wih Coningen Porfolio Programming wih he model presened in his paper has hree assumpions ha resric he generaliy of he model:. The spo price movemens are considered independen of he demand uncerainies.. I is assumed ha forward conracs exis for of any size and for any ime period. 3. The forward conracs of excess capaciy are wased in he expiraion period The firs assumpion is relevan as i gives he possibiliy o calculae he probabiliy of cerain scenarios. On he oher hand i resrics he approach o smaller marke players. This is because 7

19 significan marke players influences on he marke price and he availabiliy of he commodiy. The assumpion also misses ou he possibiliies of excepional changes, such as disasers. Togeher wih he assumpion ha all he demand can be me hese wo assumpions limi he usabiliy of he model. Again we could produce he sae ree by manually placing probabiliies, prices and demands on each sae, so ha we could ackle he issue. The second assumpion does no limi he model in mos of he cases. Whenever a large number of marke players produce commodiy, here usually is possibiliy o negoiae any kind of forward conracs (assuming ha his does no affec he price). Bu i migh be ha he forward conracs can only be sold and bough a regulaed markes and hus i may be ha his assumpion will no hold in all cases. The hird assumpion makes his model o describe he wors case scenario for he DM. If he DM could sell he forward conracs he would no lose all he invesmens in forward conracs and his would lead o a soluion where he amoun of demand secured by forward conracs could be larger. This assumpion does no limi he use of he model in mahemaical sense bu i limis he use of he model in real world cases. In some cases, we could assume ha forward conracs could be sold a heir expiraion dae a he spo price. If he ime periods are long, his is no a bad assumpion, bu i requires volaile markes. Allowing selling of fuures a a spo price on expire dae could creae problems associaed wih he ime i akes o sell he conrac and on he possible value of he conrac if par of i has already expired. However, he selling can be easily applied o he model by allowing negaive buying on spo price in saes o represen he selling of commodiy on spo price. The model presened in his paper can no be used o manage sorable commodiies. If commodiy can be sored he DM can always buy commodiy eiher using forward conracs or direcly from he marke wih spo price and sorage he goods o mee he fuure demand, depending which is cheaper. The model can be applied o any perishable commodiy ha oherwise follows he assumpions and expecaions saed in he paper. The following are examples of on wha he model can be used. Elecriciy conracs can easily be approached wih he model discussed in his paper. According o 8

20 Pirilä (005), elecriciy fuures in he Nordpool are only for cerain ime inervals. There are fuures on daily average prices for he nex 3-9 days, average prices of he nex eigh weeks, and hen here are some forward conracs. Wih hese fuures we can use his model for shor erm scenarios using days as ime periods. Wih longer scenarios we have o ake ino accoun ha here is no always possibiliy o buy fuures from any ime period o all succeeding ime periods. If we would like o use his model, we could use OTC markes o complemen he official Nordpool fuures. Bu according o Pirilä (005), mos of he OTC rading is done on official Nordpool fuures and non official fuures migh have differen coss han ha can be derived using normal fuure price formula. Alhough his model assumes ha DM does no have own producion capaciy, i could be possible o assume own producion capaciy as fuures conracs for all he ime periods (Pirilä 003). They would differ from oher fuures because heir cos would be based on he producion cos and because we would no have o use he own producion if elecriciy spo price would be less han he producion cos. Here is considered mainly he applicaion of he model wih Nordpool daa, bu basically only hings ha differ in oher exchanges are he se of available fuures and heir delivery periods (and of course heir liquidiy and possible OTC rading). In elecriciy rading, i should be noed ha here acually is very lile poin in aking shorer han one day ime periods. Balasko (00) discusses he issues relaing o differen pricing for off-peak and peak periods, which in our formula would produce wrong fuures prices if we would buy fuures from off-peak o peak, or vice versa. On he oher hand, according o Collins (00), elecriciy fuures for long ime periods are no accurae eiher. Collins examined daa from 0 fuures conracs and spo price for 3 mohs on NYSE and found ou, ha while he average variaion of he daily selemen prices for he fuures conracs was 0.03, he average variaion for he spo price was As our model penalizes for variabiliy, his indicaes ha during longer periods, hedging would probably be inense and hen probably some of i would be sold as he ime period comes closer. Oher applicaion of his model could be in he carbon dioxide (CO ) business. There are wo unknown facors, he price of CO emission (he emission exchange has boh he spo price and he forward conrac price for i) and he amoun ha he company produces emission each year. This is of course dependan on he amoun of demand and hus he revenues, bu he approach is he same. A sligh assumpion in his model is ha he CO credis ha he producer has is no aken ino 9

21 accoun (o be more general here can be assumed ha hose iniial credis assigned o ha cerain producer is so insignifican amoun ha i can be lef ou of consideraion. This, anyhow, is no always he case). Anoher variaion comes from penalizing he demand ha is no me. According o Kruger and Pizer (004), he US SO cap-and-rade program has been he basis for EU Emissions Trading Sysem, so before modifying his model o mee emission markes reader should firs sudy he exising lieraure on US SO rading program. Even logisics can be modelled wih his model. The model permis he DM o choose he amoun of logisics bough before hand. Similä (005) has used CPP o model vehicle rouing problem, where he problem is o issue he flee o cerain nodes. Maybe combining hese wo models could produce a model o describe he size of he flee as well as how o allocae i. Finally, when considering fuure research o expand he possible applicaions of he model, we noe ha he hree previously saed assumpions limi he usabiliy of he model. So finding ways around hese assumpions could produce a much more robus model. One exension for he model is no o have he demand consrains, insead penalize if he demand is no me. In a real world, in many insances, we always have he alernaive cos. I is never advisable for anyone o do anyhing ha has coss which exceed he coss of alernaive opions. The model could be exended in he sense of no having he assumpion of spo price being independen of he demand uncerainies and on how o consruc logical sae rees ha are no auomaically generaed, i.e. how each sae could have is properies independenly chosen. This would make he model much more usable in real world relaed issues. The mos imporan fuure research area is o focus on he cases in which he DM has own producion capaciy as hen he model could mee he demand of real DMs in he field of such markes where he model could prove o be usable. As presened above, he mos obvious way o hink his issue could be hrough he use of he fuures conracs. 0

22 References Gusafsson, J. and Salo, A. (005): Coningen Porfolio Programming for he Managemen of Risky Projecs, Forhcoming in Operaions Research, Sysems Analysis Laboraory, Helsinki Universiy of Technology Luenberger, D. (998): Invesmen Science, Oxford Universiy Press, New York, NY Brealey, R. and Myers, S. (003): Principles of Corporae Finance, McGraw-Hill, New York, NY Salo, A. (005): A discussion paper on forward conrac hedging using Coningen Porfolio Programming, Sysems Analysis Laboraory, Helsinki Universiy of Technology Pirilä, P. (005): Energiaalous energiamarkkina, lecure noes, Laboraory of Energy Economics and Power Plan Engineering, Helsinki Universiy of Technology Pirilä, P. (003): Energiaalous kannaavuuslaskena, lecure noes, Laboraory of Energy Economics and Power Plan Engineering, Helsinki Universiy of Technology Collins, R. (00): The Economics of Elecriciy Hedging and a Proposed Modificaion for he Fuures Conrac for Elecriciy, IEEE Transacions on Power Sysems, Vol. 7, No, February 00 Balasko, Y. (00): Theoreical perspecives on hree issues of elecriciy economics, Discussion Paper, CERAS, Paris Kruger, J. and Pizer, W. (004): The EU Emissions Trading Direcive Opporuniies and Poenial pifalls, Discussion Paper, RFF, Washingon, DC Similä, L. (005): Muli-Vehicle Rouing wih Coningen Porfolio Programming, Independen research projecs in applied mahemaics, Sysems Analysis Laboraory, Helsinki Universiy of Technology